We describe a hierarchical data analysis pipeline for coherently searching for gravitational-wave signals from non-spinning compact binary coalescences (CBCs) in the data of multiple earth-based detectors. This search assumes no prior information on the sky position of the source or the time of occurrence of its transient signals and, hence, is termed 'blind'. The pipeline computes the coherent network search statistic that is optimal in stationary, Gaussian noise. More importantly, it allows for the computation of a suite of alternative multi-detector coherent search statistics and signal-based discriminators that can improve the performance of CBC searches in real data, which can be both non-stationary and non-Gaussian. Also, unlike the coincident multi-detector search statistics that have been employed so far, the coherent statistics are different in the sense that they check for the consistency of the signal amplitudes and phases in the different detectors with their different orientations and with the signal arrival times in them. Since the computation of coherent statistics entails searching in the sky, it is more expensive than that of the coincident statistics that do not require it. To reduce computational costs, the first stage of the hierarchical pipeline constructs coincidences of triggers from the multiple interferometers, by requiring their proximity in time and component masses. The second stage follows up on these coincident triggers by computing the coherent statistics. Here, we compare the performances of this hierarchical pipeline with and without the second (or coherent) stage in Gaussian noise. Although introducing hierarchy can be expected to cause some degradation in the detection efficiency compared to that of a single-stage coherent pipeline, nevertheless it improves the computational speed of the search considerably. The two main results of this work are as follows: (1) the performance of the hierarchical coherent pipeline on Gaussian data is shown to be better than the pipeline with just the coincident stage; (2) the three-site network of LIGO detectors, in Hanford and Livingston (USA), and Virgo detector in Cascina (Italy) cannot resolve the polarization of waves arriving from certain parts of the sky. This can cause the three-site coherent statistic at those sky positions to become singular. Regularized versions of the statistic can avoid that problem, but can be expected to be sub-optimal. The aforementioned improvement in the pipeline's performance due to the coherent stage is in spite of this handicap.